An Lp-theory for a class of non-local elliptic equations related to nonsymmetric measurable kernels

We study the integro-differential operators L with kernels K(y)=a(y)J(y), where J(y) is rotationally invariant and J(y)dy is a Lévy measure on Rd (i.e. ∫Rd(1∧|y|2)J(y)dy<∞) and a(y) is an only measurable function with positive lower and upper bounds. Under few additional conditions on J(y), we prove the unique solvability of the equation Lu−λu=f in Lp-spaces and present some Lp-estimates of the solutions.